STONE DUALITY, TOPOLOGICAL ALGEBRA, AND RECOGNITION

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1 STONE DUALITY, TOPOLOGICAL ALGEBRA, AND RECOGNITION Mai Gehrke To cite this version: Mai Gehrke. STONE DUALITY, TOPOLOGICAL ALGEBRA, AND RECOGNITION. Journal of Pure and Applied Algebra, Elsevier, <hal v4> HAL Id: hal Submitted on 2 Dec 2015 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

2 STONE DUALITY, TOPOLOGICAL ALGEBRA, AND RECOGNITION MAI GEHRKE Abstract. Our main result is that any topological algebra based on a Boolean space is the extended Stone dual space of a certain associated Boolean algebra with additional operations. A particular case of this result is that the profinite completion of any abstract algebra is the extended Stone dual space of the Boolean algebra of recognisable subsets of the abstract algebra endowed with certain residuation operations. These results identify a connection between topological algebra as applied in algebra and Stone duality as applied in logic, and show that the notion of recognition originating in computer science is intrinsic to profinite completion in mathematics in general. This connection underlies a number of recent results in automata theory including a generalisation of Eilenberg-Reiterman theory for regular languages and a new notion of compact recognition applying beyond the setting of regular languages. The purpose of this paper is to give the underlying duality theoretic result in its general form. Further we illustrate the power of the result by providing a few applications in topological algebra and language theory. In particular, we give a simple proof of the fact that any topological algebra quotient of a profinite algebra which is again based on a Boolean space is again profinite and we derive the conditions dual to the ones of the original Eilenberg theorem in a fully modular manner. We cast our results in the setting of extended Priestley duality for distributive lattices with additional operations as some classes of languages of interest in automata theory fail to be closed under complementation. 1. Introduction In 1936, M. H. Stone initiated duality theory in logic by presenting a dual category equivalence between the category of Boolean algebras and the category of compact Hausdorff spaces having a basis of clopen sets, so-called Boolean spaces [55]. Stone s duality and its variants are central in making the link between syntactical and semantic approaches to logic. Also in theoretical computer science this link is central as the two sides correspond to specification languages and the space of computational states. This ability to translate faithfully between algebraic specification and spatial dynamics has often proved itself to be a powerful theoretical tool as well as a handle for making practical problems decidable. One may 2010 Mathematics Subject Classification. Primary 06D50, 20M35, 22A30; keywords: Stone/Priestley duality for lattices with additional operations, topological algebra, automata and recognition, profinite completion. This project has received funding from the European Research Council (ERC) under the European Union s Horizon 2020 research and innovation programme (grant agreement No ).. 1

3 2 MAI GEHRKE specifically mention Abramsky s paper [1] linking program logic and domain theory via Stone duality, Esakia s duality [21] for Heyting algebras and the corresponding frame semantics for intuitionistic logic, and Goldblatt s paper [31] identifying extended Stone duality as the setting for completeness issues for Kripke semantics in modal logic. These applications need more than just basic Stone duality as the first requires Stone or Priestley duality for distributive lattices and the latter two require a duality for Boolean algebras or distributive lattices endowed with additional operations. Dualities for additional operations originate with Jónsson and Tarski [34, 35] and a purely duality theoretic general account in the setting of Priestley duality may be found in [31]. Stone or Priestley duality for Boolean algebras and distributive lattices with various kinds of additional operations are often referred to as extended duality. Profinite algebragoes back at least to the paper [8] of Garrett Birkhoff, where he introduces topologies defined by congruences on abstract algebras observing that, if each congruence has finite index, then the completion of the topological algebra is compact. Profinite topologies for free groups were subsequently explored by M. Hall [32]. The profinite approach has also been used to much profit in semigroup theory and in automata theory since the late 1980s, in particular by Almeida, who developed the theory of so-called implicit operations [3]. The abstract approach to formal languages and automata provided by profinite algebra has lead to the solution of very concrete problems in automata theory, like the filtration problem [5] and the characterisation of languages recognised by reversible automata [41]. Recognisability is an original subject of computer science. Relying on automata, the notion was first introduced for finite words by Kleene [36], but was soon extended to infinite words by Büchi [15], and then further to general algebras [39], finite and infinite trees [20, 59, 49], and to many other structures. New settings in which recognition is a fruitful concept are still being developed, for example cost functions [16] and data monoids [10]. The success of the concept of recognisability has been greatly augmented by its combination with profinite methods. Our main result is a link between topological algebras based on Boolean spaces and extended Stone duality, two distinct applications of topological methods in algebra. In particular, we show that topological algebras based on Boolean spaces are always themselves dual spaces of certain Boolean algebras with additional operations. This is somewhat surprising from the point of view of duality theory as an algebraic operation f: X... X X on the dual space of an algebra A should yield coalgebra structure on the algebra in the form of h: A A... A where is coproduct(which is not an easy construction to deal with for lattices and Boolean algebras). While this is of course true, what we show here is that we actually can obtain a duality between algebras and algebras. The bulk of the paper studies this duality connecting topological algebras based on Boolean and Priestley spaces and certain Boolean algebras and distributive lattices with additional operations in detail. In particular, we identify the dual class of Boolean algebras with additional operations, the correspondence for morphisms, and the generalisation to Priestley topological algebras and their distributive lattice with additional operations duals. In the special case of the profinite completion of an algebra of any operational type, the dual Boolean algebra with additional operations is the algebra of recognisable subsets of the original algebra endowed with certain operations. This result makes clear that the use in tandem of profinite completions and recognisable subsets in

4 STONE DUALITY, TOPOLOGICAL ALGEBRA, AND RECOGNITION 3 automata theory is not accidental. Since the two are duals of each other, the study of recognisable subsets is natural, not just in automata theory and theoretical computer science, but in any setting where profinite completions occur and vice versa. The fact that the profinite completion of the free monoid on a finite set of generators is the dual space of a Boolean algebra with additional operations based on the recognisable subsets of the free monoid underlies a number of recent results in automata theory including a generalisation of Eilenberg-Reiterman theory for regular languages [26] and a notion of compact recognition applying beyond the setting of regular languages [27]. The paper is organised as follows. In Section 2 we provide the required preliminaries on duality theory. This material is not available in the needed form and with a uniform presentation, so we go in some detail. We include the discrete duality due essentially to Birkhoff as it underlies the topological one and is especially important for understanding additional operations. We describe the correspondences across the discrete and topological dualities for homomorphisms, subalgebras, and additional operations with some meet or join preservation properties. Section 3 contains the main results of the paper. We show that topological algebras over Priestley spaces are dual spaces of certain distributive lattices with additional operations, and we identify the special features of the objects on either side of the duality. Finally we consider duality for maps. In particular, we identify the dual notion to one topological algebra over a Priestley space being an (ordered) topological algebra quotient of another. This gives rise to the notion of residuation ideal. Profinite algebras are particular topological algebras based on Boolean spaces. In Section 4 we identify the lattices with additional operations dual to profinite algebras and use this characterisation to prove that Boolean-topological quotients of profinite algebras are again profinite. Then we specialise further and consider those profinite algebras which are profinite completions. In particular we show that the profinite completion of any discrete abstract algebra is the dual space of the Boolean algebra of recognisable subsets of the original abstract algebra equipped with certain residuation operations. Our proof of this result uses the general results of Section 3 and is more conceptually transparentthan the one used in [26] (see also Lemma 1 of [23]). Finally, we show how Eilenberg-Reiterman theory comes about from the duality between sublattices and quotient spaces applied in this setting. Most of the results of this paper as well as their proofs were first discovered using an algebraic approach to duality for lattices with additional operations know as the theory of canonical extensions [28]. However, in order to make the paper accessible to researchers only familiar with duality theory in its topological form, we have chosen to present the results and their proofs without reference to canonical extensions. This has the drawback that it is less transparent how we arrived at the right notions and statements of results. For an outline of the canonical extension approach to this material, see [23]. 2. Preliminaries on duality In this section we collect the basic facts about duality and extended duality that we will need. We assume all lattices to be distributive and bounded with the least element denoted by 0 and greatest element by Discrete duality. The starting point of the representation theory of distributive lattices is the classical theorem of Birkhoff for finite distributive lattices. Also,

5 4 MAI GEHRKE duality for additional operations in the infinite topological setting is obtained by adding topological requirements to the underlying discrete duality. For this reason it is interesting to review here this discrete duality generalising Birkhoff. An element p in a lattice is called join-irreducible provided p 0 and whenever p = a b, we have p = a or p = b. Theorem 2.1 (Birkhoff). Any finite distributive lattice D is isomorphic to the lattice of down-sets of the partially ordered set of join-irreducible elements of D via the assignment for a D a â := a J(D) = {p D p join-irreducible,p a}. Birkhoff s duality generalises to the category of complete lattices that are isomorphic to down-set lattices of posets. In the tradition of [34, 28], we call these DL + s. These lattices have a number of different abstract characterisations. They are the completely distributive complete lattices in which every element is the supremum of completely join-irreducible elements. Here, an element p in a complete lattice C is called completely join-irreducible provided, p = S with S C implies p S and we denote the set of all completely join-irreducible elements of C by J (C). The DL + s are also the doubly algebraic distributive lattices, see e.g. [17, p. 83] for an early textbook source. Finally, this class of lattices was also rediscovered in the domain theory community where they are known as the prime algebraic distributive lattices [40]. The Boolean members are the complete and atomic Boolean algebras, often denoted in the literature as CABAs or BA + s. Theorem 2.2. [50] Any DL + is isomorphic to the lattice of down-sets of the partially ordered set of its completely join-irreducible elements. In particular, a complete and atomic Boolean algebra is isomorphic to the powerset of its set of atoms. This correspondence between DL + s and posets extends to a categorical duality in which complete lattice homomorphisms correspond to order-preserving maps. The correspondence between complete homomorphisms h : C C and orderpreserving maps ϕ : X X is given by the following adjunction property for x X = J (C ) and u C ϕ(x ) u x h(u). This works because h has a lower adjoint which maps completely join-irreducibles to completely join-irreducibles and because h may be recovered from this map. For further details, see Section 1.1 of [24]. Let C be a DL + and X its poset of completely join irreducibles. Consider the following relation R between elements a C and pairs (x,x ) X X: a R (x,x ) (x a x a). From this relation we get a Galois connection [9] between the powersets of C and X X given by E : P(C) P(X X) : S {a (x,x ) (a R (x,x ))} K {(x,x ) a K (a R (x,x ))}

6 STONE DUALITY, TOPOLOGICAL ALGEBRA, AND RECOGNITION 5 Theorem 2.3. Let C be a DL + and X its poset of completely join irreducibles. Further, let E : P(C) P(X X) : S be the above Galois connection. The Galois closed sets are the complete sublattices of C and the quasi-orders on X extending the partial order of X, respectively. This is a poset theoretic generalisation of the correspondence between complete Boolean subalgebras of power sets and equivalence relations on the underlying sets. In Section 2.3 we will derive most of the corresponding result of topological duality (cf. Theorem 2.9). AnoperationonaDL +, f : C n C, isacomplete operator provideditpreserves arbitrary joins in each coordinate. For such an operation we have for each u C n f(u) = {f(x) x X n with x u} where X is the poset of completely join irreducible elements of C. Define R f for x X n and x X, by xr f x f(x) x. One may observe that the relations thus obtained are order-compatible in the following sense. Definition 2.4. Let X be a poset and R X n X. We say that R is ordercompatible provided for all x,x X n and all x,x X, if x x and xrx and x x, then x Rx. Remark 2.5. As we will see in Theorem 2.6, an order-compatible relation R X n X is dual to a complete operator f : C n C on the dual downset lattice C obtained from X. Since duality is contravariant it would be more natural to view R as defined above as a relation going from X to X n and this is the order of components usually used in duality theory. Also, often, the duality is described using lattices of upsets rather than downsets in order to fit with the conventional specialisation order in topology. We use downsets, as this fits better with the discrete duality of Birkhoff, and in this paper we have chosen to consider R as a relation from X n to X because we will consider many possible duals of a given relation (given by different choices of the output coordinate) and in the order chosen here R will turn out to be an operation in cases central to the theory presented in this paper. Order-compatible relations as used in topological duality theory provide a natural order-enriched generalised notion of morphisms and thus also appear in the category theoretic literature under many names (e.g. as profunctors, distributors, bimodules, order ideals). There is no general agreement there either about the order of arguments or about which coordinates should have the order reversed. For a precise connection with the category theoretic treatment of these relations, the notion of order-compatible relation given here is a distributor from X n to X, see e.g. [58] and references therein. One obtains the following discrete duality theorem for complete operators. Theorem 2.6. [35] Let C be a DL + and X its poset of completely join irreducibles. Discrete duality yields a one-to-one correspondence between the complete n-ary operators on C and the order-compatible (n+1)-ary relations on X. It is given by f : C n C R f = {(x,x) x f(x)} R X n+1 f R : D(X) n D(X), with f R (U) = R[U 1,...,U n, ]

7 6 MAI GEHRKE where R[U 1,...,U n, ] = {x X x i U i,i = 1,...,n, with (x 1,...,x n,x) R}. It is well known that an operation on a complete lattice, f : C n C, is completelyjoin-preservingintheithcoordinateifandonlyifithasanithupperresidual f # i : C n C. That is, f and f # i are related by ( ) a 1,...,a n,a C f(a 1,...,a n ) a a i f # i (a 1,...,a i 1,a i+1,...,a n,a) Also, these two maps uniquely determine each other and the fact that f # i has a lower residual is equivalent to the fact that it turns arbitrary meets in the last coordinate into meets. If, in addition, f is completely join preserving in each of its other coordinates, then f # i also turns arbitrary joins in each of the first n 1 coordinates into meets. The relation R dual to a complete operator f may also be seen as the dual of the upper residuals of f. The ith residual is given on the down-set lattice of X by (U 1,...,U i 1,U i+1,...,u n,u) (R[U 1,...,U i 1,,U i+1,...,u n,u c ]) c where ( ) c stands for the set-theoretic complement. For more details on residuation see Section 4 of [30]. The binary case is also discussed further in Section 2.4 below Ideals and filters. The basic idea of lattice duality is to represent a lattice by its set of join- and/or meet-irreducible elements. However, for infinite lattices, there aren t necessarily enough of these, and idealised elements, in the form of ideals or filters, and topology must be considered. Let D be a bounded distributive lattice. A subset I of D is an ideal provided it is a down-set closed under finite joins. We denote by Idl(D) the set of all ideals of D partially ordered by inclusion. A subset F of D is a filter provided it is an up-set closed under finite meets. Filters represent (possibly non-existing) infima and thus the order on filters is given by reverse inclusion. We denote by Filt(D) the partially ordered set of all filters of D. A proper ideal I is prime provided a b I implies a I or b I. A proper filter F is prime provided a b F implies a F orb F. Note that a filter is primeif and onlyif its complement is an ideal, which is then necessarily prime, so that prime filters and prime ideals come in complementary pairs. In particular this means that the set of prime ideals with the inclusion order is isomorphic to the set of prime filters with the reverse inclusion order. For a bounded distributive lattice D we will denote this partially ordered set by X D or just X. Since there are so many set theoretic levels in use when one talks about duality, we will revert to lower case letters x,y,z... for elements of X and to make clear when we talk about the corresponding prime filter or the complementary prime ideal we will denote these by F x and I x, respectively Stone and Priestley duality. For any bounded distributive lattice D the following map is a bounded lattice homomorphism η D : D P(X D ) a η D (a) = {x X D a F x }. 1 1 For binary operations with infix notation, we denote the two upper residuals as right and left division, see e.g. Section 2.4.

8 STONE DUALITY, TOPOLOGICAL ALGEBRA, AND RECOGNITION 7 Using the Axiom of Choice one may in addition show that D has enough prime filters/ideals in the sense that this map also is injective. The Stone dual space [56] of D is the topological space (X D,σ) where σ is the topology on X D generated by the image of the map η D, that is, by the basis {η D (a) a D}. For a Boolean algebra this yields a compact Hausdorff space for which the above basis is precisely the collection of clopen subsets of the space. For a non-boolean bounded distributive lattice the corresponding Stone space is not T 1 separated and its specialisation order is given by inclusion on the prime filters. The later Priestley variant of Stone duality [48] relies on the fact that every bounded distributive lattice,d, hasauniquebooleanextension,d, whoseprimefiltersareinone-to-one correspondence with the prime filters of D and that may be obtained by generating abooleansubalgebraofp(x D )with theimageofη D. ThusthePriestleydualspace ofaboundeddistributivelatticed istheorderedtopologicalspace(x D,,π)where x y F x F y I x I y and π is the topology on X D generated by the subbasis {η D (a),(η D (a)) c a D}. InthecasewherethelatticeDisaBooleanalgebra,thePriestleydualityagreeswith the original Stone duality for Boolean algebras [55] and we may refer to it as Stone duality rather than as Priestley duality. The dual of a homomorphism h : D E between distributive lattices in Priestley duality (as well as in Stone duality) is the map f : X E X D such that f(x) = y if and only if h 1 (F x ) = F y. One can then show that the space (X D,,π) is compact and totally order disconnected, that is, for x,y X D with x y there is a clopen down-set U with y U and x U. Also, for any homomorphism h : D E, the map h 1 : X E X D is continuous and order preserving. A Priestley space is an ordered topological space that is compact and totally order disconnected and the morphisms of Priestley spaces are the order preserving continuousmaps. ThedualofaPriestleyspace(X,,π)istheboundeddistributive lattice ClopD(X,,π) of all subsets of X that are simultaneously clopen and are down-sets. For ϕ : X Y a morphism of Priestley spaces, the restriction of the inverse image map to clopen down-sets, ϕ 1 : ClopD(Y) ClopD(X), is a bounded lattice homomorphism and is the dual of ϕ under Priestley duality. The translations back and forth given above account for Priestley duality. It allows one to translate essentially all structure, concepts, and problems back and forth between the two sides of the duality. One particular case of this translation across the duality is the correspondence between bounded sublattices of a lattice and the Priestley quotients of the dual space of the lattice. This is central to this work, and, while it is well known to duality theorists, we will supply some details here. Let i : A B be an inclusion of bounded distributive lattices. Its dual is a quotient map X B X A where x X B is sent to the point of the dual of A correspondingtotheprimefilteri 1 (F x ) = F x A. Thatis, intermsofprimefilters, the quotient map is given by restricting the prime filters of B to A. The kernel of this quotient map is a quasiordercontaining the order on X B. One can characterise the quasiorders arising in this way and this describes the correspondence. However,

9 8 MAI GEHRKE we can get something a bit better, namely, a Galois connection whose Galois closed sets are the bounded sublattices on one side and the appropriate quasiorders on the other. Let B be bounded distributive lattice and S a subset of B. Then S gives rise to a binary relation on X B given by x S y a S (a F y a F x ). It is easy to verify that S is a quasiorder extending the order on X B. In the other direction, given a subset E X B X B, we obtain a subset A E of B given by A E = {a B (x,y) E (a F y a F x )}. Here again it is easy to show that, for any E X B X B, the set A E is a bounded sublattice of B. The key facts are the following. Proposition 2.7. Let B be a bounded distributive lattice and let A be a bounded sublattice of B. Then we have A A = A. Proof. Let a 0 A and suppose x A y, that is, (x,y) A. Then by definition of A, if a 0 F y it follows that a 0 F x and thus a 0 A A. Conversely, let b A A. Fix x X B with b F x. For each y X B with b F y we then must have x A y since b A A. Thus there is a y A with a y F y but a y F x. Now we have η B (b) = {y X B b F y } {η B (a y ) y X B and b F y }. By compactness of η B (b), it follows that there are y 1,...,y n X B with η B (b) n i=1 η B(a yi ). Let a x = n i=1 a y i, then the following are true: b a x since η B is a lattice embedding and a x A since each of the a y s are in A and A is closed under finite joins. Also a x F x since F x is prime and a y F x for each y. So for each x X B with b F x, we have a x A with b a x and a x F x. The two latter facts correspond to x (η B (a x )) c (η B (b)) c. Thus we have (η B (b)) c = {(η B (a x )) c x X B and b F x }. Again, by compactness, there must be x 1,...,x m X B with b F xj for each j and m (η B (b)) c = (η B (a xj )) c. j=1 That is, b = m j=1 a x j and thus b A since A is closed under finite meets and each a x A. Further, it is easy to see that the quasiorders of the form A have the following characteristic property which we call compatibility. Definition 2.8. Let B be a bounded distributive lattice, X B the dual space of B. A quasiorder on X B is said to be compatible provided it satisfies x,y X B [x y a B (a F y and a F x and η B (a) is a -down-set)]. It is straight forward to show that A = for any compatible quasiorder on the dual of a bounded distributive lattice as compatibility easily implies that the corresponding quotient space is a Priestley space. Note that the assignments E A E and S S are both derived from the relation (x,y) R a defined by a F y a F x.

10 STONE DUALITY, TOPOLOGICAL ALGEBRA, AND RECOGNITION 9 and thus they form a Galois connection. To sum up we have the following result. Theorem 2.9 ([53]). Let B be a bounded distributive lattice, X B the dual space of B. The assignments for E X B X B and E A E = {a B (x,y) E (a F y a F x )} S S = {(x,y) X B X B a S (a F y a F x )} for S B establish a Galois connection whose Galois closed sets are the compatible quasiorders and the bounded sublattices, respectively. We note that, throughout, the special case of Stone duality for Boolean algebras corresponds to the case where the order is trivial. Remark We also note that the Priestley space of a distributive lattice is actually the dual space of the free Boolean extension D of D equipped with the compatible (quasi)order (which happens to be a partial order in this case, see [25, Proposition 8]) dual to the sublattice inclusion map D D. For more details on this, see [25, Theorem 5] Extended Priestley duality. In extended Priestley duality [31], additional operations on a distributive lattice are captured by additional relational structure on the dual space, see also [29, 30] for a description based on canonical extensions, and [33] for a different approach based more directly on category theoretic concepts. Here we give a brief description of the relational dual of the additional operations we will be most concerned with. We illustrate with a binary operation but corresponding results hold for operations of any arity. It is easiest to start with an operation : D D D preserving finite (including empty) joins in each coordinate. If D is finite, as in the discrete duality setting, it is enough to know the operation on pairs of join-irreducible elements. In the setting of arbitrary bounded distributive lattices this corresponds to knowing the action of the operation on the prime filters. For this purpose we extend the operation to an operation on the filter lattice in the obvious way: Filt : Filt(D) Filt(D) Filt(D) (F,G) F G Filt where F G Filt = { n i=1 (a i b i ) a i F and b i G,i = 1,...,n} is the filter generated by the product of F and G. The operation on filters will not in general map pairs of prime filters to prime filters but the restriction of the operation to pairs of prime filters may be encoded by the relation R = {(x,y,z) (X D ) 3 F x Filt F y F z } = {(x,y,z) (X D ) 3 F x F y F z }. In the case where the original operation preserves finite joins in each coordinate one can show that one recovers the original operation as ClopD(X D ) 2 ClopD(X D ) (U,V) R [U,V, ] = {z X D x U,y V R (x,y,z)}. Further, it may be shown that the relations R corresponding to binary operations that preserve finite joins in each coordinate are the ones satisfying the following three properties [31]: (Notice that our last coordinate is the first coordinate in [31])

11 10 MAI GEHRKE (1) ( ) R = R; (2) For each x X the set R[,,x] is closed; (3) For all U,V clopen down-sets of X the set R[U,V, ] is clopen. For operations with other preservation properties one has to apply some order duality (that is, turn the lattice upside-down). For this to work it is important that all domain coordinates transform to joins in the codomain or all transform to meets. For example, for an operation \ : D D D that sends finite joins in the first coordinate to finite meets and finite meets in the second coordinate to finite meets (when one fixes the other coordinate), we must first extend \ to a function from Filt(D) Idl(D) into Idl(D) by setting F\I = a\b a F and b I Idl for F Filt(D) and I Idl(D). The relation dual to \ is then S \ = {(x,y,z) X 3 F x \I z I y } = {(x,y,z) X 3 F x \I z I y }. and the original operation \ is captured on clopen down-sets by U\V = (S \ [U,,V c ]) c = {y x,z [(x U and S \ (x,y,z)) = z V]}. Furthermore, a relation S is the dual of some operation \ which sends finite joins in the first coordinate to finite meets and finite meets in the second coordinate to finite meets if and only if it satisfies the following three properties: (1) ( ) S = S; (2) For each x X the set S[,x, ] is closed; (3) For all U clopen down-set of X and V clopen up-set of X, the set S[U,,V] is clopen. In the sequel we will be applying these results in a situation where we have a familyofoperations(,\,/) 2 thatformaresiduatedfamilyonaboundeddistributive lattice D. That is, for all a,b,c D we have a b c b a\c a c/b. In this case one can prove that all three operations are encoded on the dual space by a single relation R which may be defined by any of the following equivalent conditions R(x,y,z) F x F y F z F x \I z I y I z /F y I x. Conversely, given a ternary relation R on a Priestley space X, which is ordercompatible so that it satisfies ( ) R = R, we obtain, via discrete duality, a residuated family of maps on the lattice of downsets of X given by S T = R[S,T, ] = {z x,y [x S and y T and R(x,y,z)]} S\T = (R[S,,T c ]) c = {y x,z [(x S and R(x,y,z)) = z T]} T/S = (R[,S,T c ]) c = {x y,z [(y S and R(x,y,z)) = z T]}. 2 Sometimes we won t have all the operations of the residuated family available on the lattice.

12 STONE DUALITY, TOPOLOGICAL ALGEBRA, AND RECOGNITION 11 However, the lattice of clopen down-sets may not be closed under some of these while being closed under others. In particular, a relation R can be the topological dual for one of these operations while not being so for another one. This is determined by the topological properties of the relation R. As stated above, this relation R is dual to an operation on D with the third coordinate as output variable if and only if (1) For each x X the set R[,,x] is closed; (2) For all U,V clopen down-sets of X the set R[U,V, ] is clopen. When this is the case we say that R is Priestley-compatible for the last coordinate. We state the topological properties for the residual operations in a definition as these are particularly central in this work and we will want to refer to them later. Definition Let X be a Priestleyspace and R X n X an order-compatible relation on X. For 1 i n, we say that R is Priestley-compatible with i as the output coordinate provided: (1) For each x X the set R[,x, ], where x occurs in the ith coordinate, is closed; (2) For all U 1,...,U i 1 and V i+1...,v n clopen down-set of X and for all W clopen up-set of X, the set R[U,,V,W] is clopen. In the setting of lattices with additional operations, we want homomorphisms to preserve both the lattice structure and the additional operations. The dual notion is known under the name of bounded morphism, see e.g. [31]. This is the functional version of bisimulation in modal logic. Definition Let X and Y be Priestley spaces, R X 3 and S Y 3 ordercompatible relations on X and Y, respectively. If R and S are Priestley-compatible with respect to the last coordinate, then we say that a continuous and orderpreserving function ϕ : X Y is a bounded morphism for these relations with respect to the last coordinate if and only if the following two properties, known as the Back and Forth properties, hold for all x 1,x 2,x 3 X and all y 1,y 2 Y (Forth) (R(x 1,x 2,x 3 ) S(ϕ(x 1 ),ϕ(x 2 ),ϕ(x 3 ))); (Back)(S(y 1,y 2,ϕ(x 3 )) z 1,z 2 [(y 1,y 2 ) (ϕ(z 1 ),ϕ(z 2 )) and R(z 1,z 2,x 3 )]). Similarly, if R X 3 and S Y 3 are Priestley-compatible with respect to the second coordinate (that is, they are duals of operations of the type \), then we say that a continuous and order-preservingfunction ϕ : X Y is a bounded morphism for these relations with respect to the second coordinate if and only if the following two properties hold for all x 1,x 2,x 3 X and all y 1,y 3 Y (Forth) (R(x 1,x 2,x 3 ) S(ϕ(x 1 ),ϕ(x 2 ),ϕ(x 3 ))); (Back) (S(y 1,ϕ(x 2 ),y 3 ) z 1,z 3 [y 1 ϕ(z 1 ),R(z 1,x 2,z 3 ), and ϕ(z 3 ) y 3 ]). If R X 3 and S Y 3 are Priestley-compatible with respect to the first coordinate (that is, they are duals of operations of the type /), then we say that a continuous and order-preserving function ϕ : X Y is a bounded morphism for these relations with respect to the first coordinate if and only if the following two properties hold for all x 1,x 2,x 3 X and all y 2,y 3 Y (Forth) (R(x 1,x 2,x 3 ) S(ϕ(x 1 ),ϕ(x 2 ),ϕ(x 3 )));

13 12 MAI GEHRKE (Back) (S(ϕ(x 1 ),y 2,y 3 ) z 2,z 3 [y 2 ϕ(z 2 ),R(x 1,z 2,z 3 ), and ϕ(z 3 ) y 3 ]). Finally, we note that in the special case where ϕ is surjective, and thus corresponds to a compatible quasiorder on X, if the quotient map is a bounded morphism for relations R on X and S on the quotient Priestley space (X/, /,π/ ) with respect to any one of their coordinates, then S is the quotient relation, R/, in the sense that for all (x,x) X n X we have ([x 1 ],...,[x n ])S[x] x[ n R ]x. 3. Topological algebras as dual spaces In this section we study the relationship between extended dual spaces and topological algebras. As we have seen in the previous section, an extended dual space is a Boolean space, or a Priestley space, with additional relations having some topological, and, in the case of Priestley spaces, order-theoretic properties. Booleantopological algebras are Boolean spaces equipped with continuous operations. Our main result in this section is that Boolean-topological algebras are precisely the extended dual spaces for which the additional relations are functional. Thus Booleantopological algebras are special extended dual spaces. In the Priestley setting the relationship is a bit more complicated: All Priestley topological algebras are extended dual spaces but they do not comprise all the functional ones. Once we have established these results, we characterise those distributive lattices with additional operations whose extended dual spaces have functional relations. Finally we identify the lattice theoretic duals of morphisms and quotients of Boolean and Priestley topological algebras. In particular we show that the dual of a Booleantopological quotient is what we will call a residuation ideal Functional dual relations. Let X be a Priestley space and R an (n+1)-ary relation on X that is order-compatible. As we have seen in the previous section, R corresponds via discrete duality to a residuated family of n-ary operations on the DL +, D(X), consisting of all down-sets of X. Depending on the topological propertiesofthisrelation, itmaythenbethedualofanynumberoftheseoperations restricted to the dual lattice. In any case, order compatibility makes it impossible for the relation to be functional unless the order is trivial (the Boolean case) or the value of the function is always minimal in X. However, given any relation R, it naturally gives rise to an order-compatible relation simply by pre- and postcomposition with the reverse order relations of its domain and codomain (this is in fact the reflection of R into the order-compatible relations). If R is the graph of an order preserving operation, then pre-composition with the reverse order relation of the domain is redundant and we obtain the following definition. Definition 3.1. Let X be a Priestleyspace and R X n X an (n+1)-aryrelation on X. We say that R is functional provided there is an n-ary operation, f, on X such that for all x X n and all z X, we have R(x,z) if and only if f(x) z. Remark 3.2. Note that if R is functional then the corresponding operation on X is uniquely given by f(x) = max{z X R(x,z)}. Further, given posets X and Y, it is not difficult to show that the assignment R R := R is a reflection of relations from X to Y into those that are order-compatible. Further, one may

14 STONE DUALITY, TOPOLOGICAL ALGEBRA, AND RECOGNITION 13 show that R is functional if and only if there exists an (order-compatible) relation R from Y to X so that R R and R R. For the category theory minded reader, we note that this latter statement may be seen as the fact that R has an right adjoint in the bicategory of distributors/profunctors over the category of posets, see e.g. [12, Propositions and 7.9.2] for a general version of this fact. The truth of the following proposition is easy to verify. Proposition 3.3. Let X be a Priestley space and R a functional relation on X with f the corresponding operation. Then f is order preserving if and only if R is order-compatible. The following establishes a link between continuity of an operation on a Priestley space and the corresponding functional relation being the topological dual of the residual operations given by the relation. Proposition 3.4. Let X be a Priestley space and R X n X an order-compatible functional relation on X with f the corresponding operation. Then the following conditions are related by (i) (ii), (ii) (iii), and (iii) (iv). (i) The operation f is continuous with respect to the Priestley topology. (ii) For each i with 1 i n and all x X, R[,x, ] (where x is in the ith spot) is closed in X n and for all clopen down-sets U j,v X the relational image R[U 1,...,U i 1,,U i+1,...,v c ] is clopen. (iii) There is an i with 1 i n such that for all x X, R[,x, ] (where x is in the ith spot) is closed in X n and for all clopen down-sets U j,v X the relational image R[U 1,...,U i 1,,U i+1,...,v c ] is clopen. (iv) The operation f is continuous with respect to the spectral topology. Proof. Assuming that (i) holds, we just prove (ii) for i = 1 to minimise notation. If f is continuous in the Priestley topology, then, for each x 1 X, the function f x1 : X n 1 X given by y f(x 1,y) is continuous in the Priestley topology and thus its graph, G(f x1 ), is closed in X n. Also notice that if X op denotes the Priestley space obtained by reversing the order of X, then R[x 1, ] is the down-set of G(f x1 ) in the order of the space (X op ) n 1 X. Now using the fact that the product of Priestley spaces is a Priestley space, and that down-sets of closed sets are closed in Priestley spaces [18, Exercise11.14(ii)], we conclude that R[x 1, ] is closed. Let U 2,...,U n,v be clopen down-sets in X. By continuity of f, the set f 1 (V c ) is clopen. Also, because V c is an up-set, f 1 (V c ) = R[,V c ]. Now let π : X X n 1 X be the projection onto the first coordinate, then R[,U 2,...,U n,v c ] = π(r[,v c ] (X U 2... U n )). Since the intersection of clopen sets is clopen, projections of open sets are open, and projections of closed sets along compact Hausdorff spaces are closed, it follows that R[,U 2,...,U n,v c ] is clopen. This completes the proof of (i) implies (ii). In order to prove that (iii) implies (iv), we assume (iii) holds for i = 1 and prove that f is continuous with respect to the spectral topology. To this end, let (x 1,...,x n ) X n with z = f(x 1,...,x n ) V, where V is a clopen down-set in X. Since R[x 1, ] is closed and V is clopen, it follows that R[x 1, ] (X n V c ) is closed. Furthermore, since the projection π : X n 1 X X n 1 is a projection

15 14 MAI GEHRKE along a compact space, it is a closed map and thus π (R[x 1, ] (X n 1 V c )) is closed. Notice that since V c is an up-set we have π (R[x 1, ] (X n 1 V c )) = {(y 2,...,y n ) X n 1 z V c f(x 1,y 2,...,y n ) z} = {(y 2,...,y n ) X n 1 f(x 1,y 2,...,y n ) V c } = f 1 x 1 (V c ) wheref x1 : X n 1 X isistherestrictionoff asdefinedabove. Itthusfollowsthat fx 1 1 (V) is open. Since it is also a down-set and x = (x 2,...,x n ) fx 1 1 (V), there are clopen down-sets U 2,...,U n with x U 2... U n fx 1 1 (V). We have x 1 R[,U 2,...,U n,v c ], and thus x 1 (R[,U 2,...,U n,v c ]) c = U 1 which is a down-set andis inaddition clopen bycondition(iii). Thatis, (x 1,...,x n ) U 1 U 2... U n and each U i is a clopen down-set. Furthermore, if (y 1,...,y n ) U 1 U 2... U n then y 1 U 1 = (R[,U 2,...,U n,v c ]) c and thus y 1 R[,U 2,...,U n,v c ] = {y (y 2,...,y n) U 2... U n f(y,y 2,...,y n) V}. That is, for all (y 2,...,y n ) U 2... U n we have f(y 1,y 2,...,y n ) V and in particular f(y 1,y 2,...,y n ) V. We have shown then that (x 1,...,x n ) U 1 U 2... U n f 1 (V) and thus that f is continuous in the spectral topology. In the Boolean case, we obtain a stronger result since the Priestley and the spectral topologies are one and the same so that conditions (i) and (iv) are equivalent. Corollary 3.5. Let X be a Boolean space and let f be an n-ary operation on X and suppose R X n X is the graph of f. Then the following conditions are equivalent: (1) There is an i with 1 i n such that R is the extended Stone dual of the operation (U 1,...,U n ) R[U 1,...,,...,U n ] (with the co-domain slot in the ith place) on the dual Boolean algebra. (2) For each i with 1 i n, the relation R is the extended Stone dual of the operation (U 1,...,U n ) R[U 1,...,,...,U n ] (with the co-domain slot in the ith place) on the dual Boolean algebra. (3) The operation f is continuous. Proposition 3.4 and its corollary allow us to relate extended dual spaces and the standard notion of topological algebras. Definition 3.6. Given an operational type τ, a topological algebra of type τ is an algebra of type τ in the category of topological spaces. That is, it is a topological space equipped with an algebraic structure of type τ for which each basic operation is continuous (in the case of an n-ary operation we equip the domain with the product topology). Homomorphisms of topological algebras are maps which are simultaneously homomorphisms for the algebra structure and continuous for the topological structure. Isomorphisms must also be homeomorphisms for the topological part of the structure. A topological algebra is said to be a Boolean-topological algebra provided the underlying topological space is a Boolean space, i.e., it is compact Hausdorff with a basis of clopen sets. Finally, a Priestley topological algebra is an algebra in the category of Priestley spaces. That is, it is a Priestley space equipped with an algebra structure such that each basic operation of the algebra

16 STONE DUALITY, TOPOLOGICAL ALGEBRA, AND RECOGNITION 15 is not only continuous but also order preserving. The homomorphisms are algebra homomorphisms that are continuous and order preserving, whereas isomorphisms also have to be homeomorphisms for the topological structure and isomorphisms for the order structure. Applying the implications (i) (ii) and/or(i) (iii), and noticing that conditions (ii) and (iii) are precisely the n-ary versions of the conditions for being dual to a residual operation given in Definition 2.11, we obtain the following corollary of Proposition 3.4. Corollary 3.7. Every Priestley topological algebra is the dual space of some bounded distributive lattice with additional operations. Corollary 3.8. The Boolean-topological algebras are precisely the extended Boolean dual spaces of Boolean algebras with residuation operations for which the dual relations are functional. Proposition 3.4 establishes a connection between the duals of residual operations and continuous maps. One may wonder what it takes for the forward image map to be the dual of an operation. Using the condition, as given in Section 2, on a relation equivalent to it being the dual of the forward image operation, we obtain the following requirements in the case of a functional relation on a Boolean space: (1) For all x X the preimage f 1 (x) is closed; (2) The forward image of a tuple of clopens is clopen. Without continuity this is not a very natural condition for a map between topological spaces. However, we do obtain the following useful corollary. Corollary 3.9. Let X be a Boolean-topological algebra, B the dual Boolean algebra, and f one of the basic operations of X. Then f is an open mapping if and only if B is closed under the forward operation (U 1,...,U n ) f[u 1... U n ], and in this case the graph R of f is also the relational dual to this forward operation on B. Proof. Since all continuous maps from compact spaces to Hausdorff spaces are closed mappings, it follows that B is closed under the operation (U 1,...,U n ) f[u 1... U n ] if and only if f is an open map. The conditions required for R to be the dual of this operations are f 1 (x) = R[,x] closed for each x X and f[u 1... U n ] = R[U 1,...,U n, ] clopen whenever the U i s are clopen. For f continuous and X Hausdorff the first condition always holds. And if in addition f is open, then the second also holds. In the remainder of this subsection we investigate the relationship between the conditions(i) (iv) of Proposition 3.4 further. In particular we show that, for Priestleyspacesin general, condition(i) isequivalent to(ii) (and (iii)) in the caseofunary operations, but that (ii) and (iii) are equivalent to neither of (i) and (iv) in general. Proposition Let X be a Priestley space and R X X an order-compatible and functional binary relation on X with f the corresponding operation. Then the following conditions are equivalent. (1) The operation f is continuous with respect to the Priestley topology. (2) For all x X, R[x, ] is closed in X and for all clopen down-sets V X the relational image R[,V c ] is clopen.

17 16 MAI GEHRKE Furthermore, if these conditions are satisfied, then the dual operation V (R[,V c ]) c is the distributive lattice homomorphism dual to f. Proof. We already know from Proposition 3.4 that (i) implies (ii). For the reverse implication, note that if V is a clopen down-set then its complement is an up-set, and for up-sets U we have R[,U] = {x X y U with f(x) y} = {x X f(x) U} = f 1 [U]. So condition (ii) implies that the preimages of clopen up-sets are clopen. Now since f 1 [U c ] = (f 1 [U]) c, the preimages of clopen down-sets are also clopen, and since the clopen up-sets and clopen down-sets together form a subbasis for the Priestley topology, it follows that (ii) implies (i) as required. Finally, we now see that the operation V (R[,V c ]) c is equal to the lattice homomorphism V f 1 [V] since, by the above computation (R[,V c ]) c = (f 1 [V c ]) c = ((f 1 [V]) c ) c = f 1 [V] for any clopen down-set V. We thus see that unary Priestley topological algebras are rather trivial, as one also expects since the dual of an order preserving continuous map under Priestley duality is a homomorphism. Corollary The unary Priestley topological algebras are precisely the extended Priestley dual spaces of distributive lattices with additional operations which are endomorphisms of the lattice. As detailed in the following example, we can also use Proposition 3.10 to show that the last condition of Proposition 3.4 is not equivalent to the first three in general. Example Consider the bounded distributive lattice D of all subsets of Z which are either finite or Z itself. The Priestley dual of D is the poset X = Z obtained by adding as a top to the trivially ordered anti-chain Z. The topology on X is the one of the one-point compactification by of the discrete space on Z. Now consider the map f : X X which sends any k Z to 0 and to. Then f is continuous in the spectral topology but not in the Priestley topology. Conditions (2) and (3) of Proposition 3.4 also are not satisfied since, by Proposition 3.10, these are equivalent to continuity in the Priestley topology in the unary case. Finally, wealsogiveanexampletoshowthatthefirstconditionofproposition3.4 is not equivalent to the last three in general. Example Consider again the bounded distributive lattice D of all subsets of Z which are either finite or Z itself. We consider the residuals of addition on Z lifted to the power set of Z. Since addition is commutative, the right and left residuals agree and we need only consider one of them. Note that for A,B P(Z), we have A/B = {A/k k B} = {A k k B}. Further, it is clear that D is closed under ( ) k as F k = {n k n F} is again finite for F finite and Z k = Z. Also, D is closed under arbitrary intersections, so D is closed under residuation. As in the previous example, the Priestley dual of D is the poset X = Z obtained by adding as a top to the trivially ordered anti-chain Z and the topology on X is the one of the one-point compactification by of the discrete space on Z. It is straight forward to verify that the ternary

18 STONE DUALITY, TOPOLOGICAL ALGEBRA, AND RECOGNITION 17 relation dual to the residuation operation on D is functional with its upper edge given by addition on X defined as usual; for i,j Z: + i j i+j Since we are dealing with the extended dual of a residuation operation, conditions (ii) and (iii), and thus also (iv) of Proposition 3.4 must be satisfied. However, this addition operation is not continuous in the Priestley topology since, e.g., the singleton {0} is clopen but its preimage, {(k, k) k Z}, is open but not closed. We conclude that in Proposition 3.4 conditions(ii) and(iii) are neither equivalent to condition (i) nor to condition (iv) in general. We postpone a characterisation of the duals of Priestley topological algebras to Section Residuation algebras preserving joins at primes. In this subsection, we characterise the additional operations on lattices for which the extended Priestley dual relations are functional. In the exposition, we will mainly focus on the binary case in order to lighten the notation. The results do go through for higher arities as well though. For a unary operation on a lattice, we saw in Proposition 3.10, that its dual relation is functional if and only if the operation is in fact an endomorphism of the lattice. However, the situation is far from this trivial in the binary and higher arity setting. In fact, in arities greater than or equal to two, a dual relation may be functional without the original map preserving both meet and join. To see this, consider the set X = {0,1, 1} with the ternary relation given by usual multiplication. The binary residuation operation / on B = P(X) preserves meet in its first coordinate and reverses joins in the second, that is, the identities (A 1 A 2 )/B = (A 1 /B) (A 2 /B) and A/(B 1 B 2 ) = (A/B 1 ) (A/B 2 ) hold in B. However, the operation / does not preserve join in the first coordinate nor does it reverse meets in the second, e.g., { 1,1}/{ 1,1} = { 1,1} but { 1}/{ 1,1} = = {1}/{ 1,1}, and {1}/ = X is strictly larger than the union of {1}/{1}= {1} and {1}/{ 1}= { 1}. As we have seen in the previous subsection, the appropriate operations on lattices dual to functional relations are residuation operations. We make the following definition. Definition A (binary) residuation algebra is a bounded distributive lattice, D, equipped with two binary operations \,/ : D D D with the following properties: (1) a,b 1,b 2 D a\1 = 1 and a\(b 1 b 2 ) = (a\b 1 ) (a\b 2 ). That is, \ preserves finitary meets in the second coordinate. (2) a 1,a 2,b D 1/b = 1 and (a 1 a 2 )/b = (a 1 /b) (a 2 /b). That is, / preserves finitary meets in the first coordinate. (3) The two operations \ and / are linked by the Galois property: a,b,c D b a\c a c/b.